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Concept
Bisection
Summary
In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a 'bisector'. The most often considered types of bisectors are the 'segment bisector' (a line that passes through the midpoint of a given segment) and the 'angle bisector' (a line that passes through the apex of an angle, that divides it into two equal angles).
In three-dimensional space, bisection is usually done by a bisecting plane, also called the 'bisector'.
The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly.
The perpendicular bisector of a line segment also has the property that each of its points is equidistant from segment AB's endpoints:
(D).
The proof follows from and Pythagoras' theorem:
Property (D) is usually used for the construction of a perpendicular bisector:
In classical geometry, the bisection is a simple compass and straightedge construction, whose possibility depends on the ability to draw arcs of equal radii and different centers:
The segment is bisected by drawing intersecting circles of equal radius , whose centers are the endpoints of the segment. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment.
Because the construction of the bisector is done without the knowledge of the segment's midpoint , the construction is used for determining as the intersection of the bisector and the line segment.
This construction is in fact used when constructing a line perpendicular to a given line at a given point : drawing a circle whose center is such that it intersects the line in two points , and the perpendicular to be constructed is the one bisecting segment .
If are the position vectors of two points , then its midpoint is and vector is a normal vector of the perpendicular line segment bisector. Hence its vector equation is . Inserting and expanding the equation leads to the vector equation
(V)
With one gets the equation in coordinate form:
(C)
Or explicitly:
(E),
where , , and .
Official source
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